Question

Solve the following inequation:
$(x-5)(x+9)(x-8)<0$

Answer 1

$(x+9)(x-5)(x-8)<0$

Let $f\left(x\right)=(x-5)(x+9)(x-8)$

Here the critical points are $-9,5,8$

Plot these points on a number line. These points divide the number line into $4$ intervals.

Now, pick a point from each interval and find the sign of $f\left(x\right)$

$\Rightarrow f(-10)<0$

$\Rightarrow f\left(0\right)>0$

$\Rightarrow f\left(6\right)<0$

$\Rightarrow f\left(9\right)>0$

Since the inequality given in the question is ${}^{\prime }{<}^{\prime }$, the solution is the interval containing ${}^{\prime }{-}^{\prime }$ sign

Thus, $x$ lies in the interval $(-\infty ,-9)\cup (5,8)$